The three-body decay \(\ell _\alpha \rightarrow \ell _\beta \bar{\ell }_\beta \ell _\beta \)  is another kind of LFV process. The current limit on BR(\(\mu \rightarrow 3e\) ) is \(1.0\times 10^{-12}\)  [1]} and is expected to reach \(10^{-16}\)  in the future [2]}. The limits on BR(\(\tau \rightarrow 3e\) ) and BR(\(\tau \rightarrow 3\mu \) ) are \(2.7\times 10^{-8}\)  and \(2.1\times 10^{-8}\)  [3]}.
Considering the simplified scenario with small Yukawa coupling, e.g. \(h^{\prime }\lesssim 0.01\) , then the three-body decay \(\ell _\alpha \rightarrow \ell _\beta \bar{\ell }_\beta \ell _\beta \)  is dominant by the \(\gamma \) -penguins. The branching ratio is given by
\(\text{BR}(\ell _\alpha \rightarrow \ell _\beta \bar{\ell }_\beta \ell _\beta ) &=&\frac{3 (4\pi )^2 \alpha _\text{em}^2}{8G_F^2}\bigg [ |A_{ND}|^2 +|A_D|^2\left(\frac{16}{3}\log \left(\frac{m_\alpha }{m_\beta }\right)-\frac{22}{3}\right) \\ \nonumber &&\hspace{50.0pt} - \big (2 A_{ND} A_D^*+\text{h.c.}\big )\bigg ]\times \text{BR}(\ell _\alpha \rightarrow \ell _\beta \nu _\alpha \bar{\nu }_\beta ),\) 
